Can more polynomial time compensate for less polynomial memory? I'm wondering what is known about the relation between time and memory for polynomial-time algorithms (which are necessarily also polynomial-space). In particular, I would like to learn what is known about less time vs. less memory, in the following sense:
Is it known whether (say) any language $L$ recognized by a polynomial-time algorithm $A$ which is polynomial-space $O(n^s)$ ($s>2$ say), can also be recognized by a polynomial-time algorithm $B$ which is polynomial-space $O(n^{s/2})$? Presumably any $B$ with less memory would need significantly more time, but can we restrict this extra time to polynomial-time? 
So, recapping: is there for poly-time algorithms, some general way to compensate for less memory, in polynomial time? Answers and references greatly appreciated.   
update: Dan Brumleve and Timothy Chow have given valuable insights and references in the comments and in this chat. It seems that time-space trade-offs like the one in this question are still a bridge too far.
 A: I suspect that what you're hoping for—i.e., that for all $a$ and $b$ there exists $c$ such that  $\mathrm{DTISP}(n^a, n^b) \subseteq \mathrm{DTISP}(n^c, n^{b/2})$—is false, but for sure nobody has proved this.  In fact I suspect it may not even be true that $\mathrm{DTISP}(n^a,n^b) \subseteq\mathrm{DSPACE}(n^{b/2})$ (i.e., I suspect you cannot compensate for the loss of space with any amount of time).  Certainly, if we consider all problems solvable in space $O(n^b)$ (with no restrictions on time), then 
the space hierarchy theorem tells us that we will be able to solve certain problems that are simply unsolvable in space $O(n^{b/2})$, no matter how much time we give ourselves.  But settling your question will probably require proving a timespace hierarchy theorem, and very little is known unconditionally about such things (see this question on CS Theory Stackexchange for some information).  As another illustration of our ignorance about time-space tradeoffs, note that it is not known whether a problem that can be solved in polynomial time, and that can also be solved (possibly by a different algorithm) in polylogarithmic space, is in SC (i.e., is solvable is polynomial time and polylogarithmic space simultaneously).
A: It is an open problem whether or not every polynomial time algorithm can be made $O(\log(n))$-space.  Note that every $O(\log(n))$-space algorithm is simultaneously polynomial time because it has $2^{O(\log(n))} = n^{O(1)}$ states.  This problem is usually referred to as $\text{P} = \text{L}$.  Amazingly, $\text{NP} = \text{L}$ is also open!
If we want to talk about particular algorithms and polynomial exponents, we'll also want to get specific about the model of computation.  For example, consider the class of models of computation related to a Turing machine by time translations satisfying $T_b \in T_a \cdot S_a^{O(1)}$ and space translations satisfying $S_b \in O(S_a)$.  This includes single-tape and multi-tape Turing machines and variations of $\text{RAM}$.  We can't say much, since an algorithm with $T_a = S_a$ in one model can have $T_b = S_b^{O(1)}$ in the other, that is, by changing the model an algorithm can be made arbitrarily more space-efficient relative to the time it takes.
I posed a related problem recently over on cstheory.SE.
